Updated Java-1.6.0-OpenJDK packages are available for CentOS:

CESA-2016:0067 Important CentOS 5 java-1.6.0-openjdk Security Update
CESA-2016:0067 Important CentOS 6 java-1.6.0-openjdk Security Update
CESA-2016:0067 Important CentOS 7 java-1.6.0-openjdk Security Update

CESA-2016:0067 Important CentOS 5 java-1.6.0-openjdk Security Update

CentOS Errata and Security Advisory 2016:0067 Important

Upstream details at : https://rhn.redhat.com/errata/RHSA-2016-0067.html

The following updated files have been uploaded and are currently
syncing to the mirrors: ( sha256sum Filename )

i386:
a564cd9490be5ab97d050c1a1cee2090f315dc6e8993c3bb57fac0c732c6a3d3 java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el5_11.i386.rpm
1e6e6102a88e6f8d1d99ba513f5d5bd27445c08ac35606126330dcb7eb309d8b java-1.6.0-openjdk-demo-1.6.0.38-1.13.10.0.el5_11.i386.rpm
5c5f8fed0f9e6ea7f009fa5db9f1d8517160a68470d5fa42d3b5964c730e1e12 java-1.6.0-openjdk-devel-1.6.0.38-1.13.10.0.el5_11.i386.rpm
e3a65df6870d94b0be2553dabb647892d44564fc25ce7f091a50e711fb1bb6a3 java-1.6.0-openjdk-javadoc-1.6.0.38-1.13.10.0.el5_11.i386.rpm
7596230198a5fc3d7149f94becde9cccccc1166f03b7d664500df33bae23ea67 java-1.6.0-openjdk-src-1.6.0.38-1.13.10.0.el5_11.i386.rpm

x86_64:
65e34e63b9a6d16a8019e1e7027d41438ae81722373d5847c94a0ee879a5478b java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el5_11.x86_64.rpm
3018d3ea3b1a291928891409ac60be0120863848ded1f271cc2b6a332a45b102 java-1.6.0-openjdk-demo-1.6.0.38-1.13.10.0.el5_11.x86_64.rpm
276a82950dbe20324ce1322cf2cd781466b8a0b8a155b6d7f7f46a0bce2e76d9 java-1.6.0-openjdk-devel-1.6.0.38-1.13.10.0.el5_11.x86_64.rpm
dc3173332668d4a3a75894c73ed938cf558592d49cc3e18c86e03a92ad776ba8 java-1.6.0-openjdk-javadoc-1.6.0.38-1.13.10.0.el5_11.x86_64.rpm
e14622d70222762fad88e2f2238e6ae27c8f630e0ed29dbd78c19e0d2d0fb329 java-1.6.0-openjdk-src-1.6.0.38-1.13.10.0.el5_11.x86_64.rpm

Source:
67daf7f23cfdf1dc5aecdd9d6e1fdaedf77d3863445da21200f52628ec412953 java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el5_11.src.rpm

CESA-2016:0067 Important CentOS 6 java-1.6.0-openjdk Security Update

CentOS Errata and Security Advisory 2016:0067 Important

Upstream details at : https://rhn.redhat.com/errata/RHSA-2016-0067.html

The following updated files have been uploaded and are currently
syncing to the mirrors: ( sha256sum Filename )

i386:
8df06a899e26a7520bdeb3b4db31a8fe4c4686e10b2fefde977664de9c7fb658 java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el6_7.i686.rpm
b645b00280c6b2df08eff4b3ea35f7c5dadffd5c261cfab385aaf9b1b9f37c39 java-1.6.0-openjdk-demo-1.6.0.38-1.13.10.0.el6_7.i686.rpm
1a3a289f87f54ac2e0b351171f969e10c18f2b7f9068b37ffb962ecf3e5f2480 java-1.6.0-openjdk-devel-1.6.0.38-1.13.10.0.el6_7.i686.rpm
bcae5c7e8520c7462c383d629d5e74047e2ce5991b7ac22f36ac035f0fc33000 java-1.6.0-openjdk-javadoc-1.6.0.38-1.13.10.0.el6_7.i686.rpm
30a96ce89d2188f1ee96309dfd9ba72f37c77d3d8cc924baf3dcd25ae786d5f7 java-1.6.0-openjdk-src-1.6.0.38-1.13.10.0.el6_7.i686.rpm

x86_64:
dc86666020c01029080793f7a80ee11ff7b8aa692ee3e5d27b36281d54f5e1a3 java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el6_7.x86_64.rpm
8090ed5219f8f569c4fb0dc94242b0fcfe62415de1089b1a99ade591102ba088 java-1.6.0-openjdk-demo-1.6.0.38-1.13.10.0.el6_7.x86_64.rpm
7f3c15ee1836066bad87abca524b4af50cc449f51f7ef5a248f4f3db2b2d62af java-1.6.0-openjdk-devel-1.6.0.38-1.13.10.0.el6_7.x86_64.rpm
4ac324c5649d25af27d90af2b6310924c029ac7063f922a43e946969051cc92b java-1.6.0-openjdk-javadoc-1.6.0.38-1.13.10.0.el6_7.x86_64.rpm
0b05e4b8e47cf5c7e6d7daf23c9eb8ab620ac85840dbc6d4d50bf5c4a4aa9743 java-1.6.0-openjdk-src-1.6.0.38-1.13.10.0.el6_7.x86_64.rpm

Source:
5caca590014371e8066406ef364651473a52b86e7fc56f33acbef9d9b9aa2d80 java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el6_7.src.rpm

CESA-2016:0067 Important CentOS 7 java-1.6.0-openjdk Security Update

CentOS Errata and Security Advisory 2016:0067 Important

Upstream details at : https://rhn.redhat.com/errata/RHSA-2016-0067.html

The following updated files have been uploaded and are currently
syncing to the mirrors: ( sha256sum Filename )

x86_64:
6d2a9fcd8c0047fbf75850d6f87916da874d826cd045fd282624d4e2c9027e2a java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el7_2.x86_64.rpm
fc340ba9fdf41498a57f1e855c5ef4d914b1bf02dda5c3532efecdb721fbb64b java-1.6.0-openjdk-demo-1.6.0.38-1.13.10.0.el7_2.x86_64.rpm
d20b7ecea5d222dd53a8795dd24ef962e6fea3e12169dabaa09408fb99c6f96f java-1.6.0-openjdk-devel-1.6.0.38-1.13.10.0.el7_2.x86_64.rpm
88daac1696331dc127050b86ead11f1e1311b630f2e8cb0105e09c54293b2a22 java-1.6.0-openjdk-javadoc-1.6.0.38-1.13.10.0.el7_2.x86_64.rpm
a2e0e4314c57c089bd5a0269b43dd4f90225f689c3cf31ade4445959c8d44c72 java-1.6.0-openjdk-src-1.6.0.38-1.13.10.0.el7_2.x86_64.rpm

Source:
93a4207025786c507681644aeb2d0b477bc7ff567c95a7275df008576d8a61dc java-1.6.0-openjdk-1.6.0.38-1.13.10.0.el7_2.src.rpm